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Derivation

1. Split input into 2 regimes

2. if x < 2.045816106302674

   1. Initial program 39.3

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   2. Taylor expanded around 0 1.2

      \[\leadsto \frac{\color{blue}{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}}}{2}\]
   3. Using strategy rm

   4. Applied unpow21.2

      \[\leadsto \frac{\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - \color{blue}{x \cdot x}}{2}\]

   5. Applied add-sqr-sqrt2.1

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} \cdot \sqrt{\frac{2}{3} \cdot {x}^{3} + 2}} - x \cdot x}{2}\]

   6. Applied difference-of-squares2.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} + x\right) \cdot \left(\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} - x\right)}}{2}\]
   7. Using strategy rm

   8. Applied flip-+2.1

      \[\leadsto \frac{\color{blue}{\frac{\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} \cdot \sqrt{\frac{2}{3} \cdot {x}^{3} + 2} - x \cdot x}{\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} - x}} \cdot \left(\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} - x\right)}{2}\]

   9. Applied associate-*l/2.1

      \[\leadsto \frac{\color{blue}{\frac{\left(\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} \cdot \sqrt{\frac{2}{3} \cdot {x}^{3} + 2} - x \cdot x\right) \cdot \left(\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} - x\right)}{\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} - x}}}{2}\]

   10. Simplified1.2

       \[\leadsto \frac{\frac{\color{blue}{\left(\sqrt{2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)} - x\right) \cdot \left(2 + \left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) - x \cdot x\right)\right)}}{\sqrt{\frac{2}{3} \cdot {x}^{3} + 2} - x}}{2}\]

   11. Taylor expanded around 0 1.2

       \[\leadsto \frac{\frac{\left(\sqrt{2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)} - x\right) \cdot \left(2 + \left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) - x \cdot x\right)\right)}{\color{blue}{\left(\left(\sqrt{2} + \frac{1}{3} \cdot \frac{{x}^{3}}{\sqrt{2}}\right) - \frac{1}{18} \cdot \frac{{x}^{6}}{{\left(\sqrt{2}\right)}^{3}}\right)} - x}}{2}\]

   if 2.045816106302674 < x

   1. Initial program 0.3

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

   2. Taylor expanded around -inf 0.3

      \[\leadsto \frac{\color{blue}{\left(\frac{e^{x \cdot \varepsilon - x}}{\varepsilon} + \left(e^{x \cdot \varepsilon - x} + e^{-\left(x \cdot \varepsilon + x\right)}\right)\right) - \frac{e^{-\left(x \cdot \varepsilon + x\right)}}{\varepsilon}}}{2}\]
3. Recombined 2 regimes into one program.

4. Final simplification1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 2.045816106302674:\\ \;\;\;\;\frac{\frac{\left(2 + \left(\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) - x \cdot x\right)\right) \cdot \left(\sqrt{2 + \left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right)} - x\right)}{\left(\left(\sqrt{2} + \frac{{x}^{3}}{\sqrt{2}} \cdot \frac{1}{3}\right) - \frac{{x}^{6}}{{\left(\sqrt{2}\right)}^{3}} \cdot \frac{1}{18}\right) - x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{e^{\varepsilon \cdot x - x}}{\varepsilon} + \left(e^{\left(-x\right) + \left(-x\right) \cdot \varepsilon} + e^{\varepsilon \cdot x - x}\right)\right) - \frac{e^{\left(-x\right) + \left(-x\right) \cdot \varepsilon}}{\varepsilon}}{2}\\ \end{array}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018348 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  (/ (- (* (+ 1 (/ 1 eps)) (exp (- (* (- 1 eps) x)))) (* (- (/ 1 eps) 1) (exp (- (* (+ 1 eps) x))))) 2))